Solve the exponential equation for $x$. 8 6 x − 5 ⋅ 2 3 x + 1 = 2 8 x + 9 8\^{6x-5}\cdot 2\^{ 3x+1}=2\^{ 8x+9} $x=$
Explanation: The strategy Let's write $8$ in base $2$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $2$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 8 6 x − 5 ⋅ 2 3 x + 1 = ( 2 3 ) 6 x − 5 ⋅ 2 3 x + 1 = 2 18 x − 15 ⋅ 2 3 x + 1 = 2 18 x − 15 + ( 3 x + 1 ) = 2 21 x − 14 ( 8 = 2 3 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 8\^{6x-5}\cdot 2\^{ 3x+1}&=(2^3)\^{6x-5}\cdot 2\^{ 3x+1}&&&&(8=2^3)\\\\ &=2\^{C{18x-15}}\cdot 2\^{ {3x+1}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=2\^{ C{18x-15} \ + \ ({3x+1}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=2\^{ 21x-14} \end{aligned} Solving the linear equation We obtain the following equation. 2 21 x − 14 = 2 8 x + 9 2\^{ 21x-14}=2\^{ 8x+9} Now we can equate the exponents and solve for $x$. $\begin{aligned} 21x-14 &=8x+9\\\\ x &= \dfrac{23}{13}\end{aligned}$ The answer The answer is $x=\dfrac{23}{13}$. You can check this answer by substituting $\it{x=\dfrac{23}{13}}$ in the original equation and evaluating both sides.